Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 10 (2014), 005, 21 pages      arXiv:1309.3713      http://dx.doi.org/10.3842/SIGMA.2014.005

Why Do the Relativistic Masses and Momenta of Faster-than-Light Particles Decrease as their Speeds Increase?

Judit X. Madarász a, Mike Stannett b and Gergely Székely a
a) Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, P.O. Box 127, Budapest 1364, Hungary
b) University of Sheffield, Department of Computer Science, 211 Portobello, Sheffield S1 4DP, United Kingdom

Received September 17, 2013, in final form January 07, 2014; Published online January 11, 2014

Abstract
It has recently been shown within a formal axiomatic framework using a definition of four-momentum based on the Stückelberg-Feynman-Sudarshan-Recami ''switching principle'' that Einstein's relativistic dynamics is logically consistent with the existence of interacting faster-than-light inertial particles. Our results here show, using only basic natural assumptions on dynamics, that this definition is the only possible way to get a consistent theory of such particles moving within the geometry of Minkowskian spacetime. We present a strictly formal proof from a streamlined axiom system that given any slow or fast inertial particle, all inertial observers agree on the value of $\mathsf{m}\cdot \sqrt{|1-v^2|}$, where $\mathsf{m}$ is the particle's relativistic mass and $v$ its speed. This confirms formally the widely held belief that the relativistic mass and momentum of a positive-mass faster-than-light particle must decrease as its speed increases.

Key words: special relativity; dynamics; faster-than-light particles; superluminal motion; tachyons; axiomatic method; first-order logic.

pdf (593 kb)   tex (204 kb)

References

  1. Aharonov Y., Erez N., Reznik B., Superoscillations and tunneling times, Phys. Rev. A 65 (2002), 052124, 5 pages, quant-ph/0110104.
  2. Andréka H., Madarász J.X., Németi I., On the logical structure of relativity theories, Research report, Alfréd Rényi Institute of Mathematics, Hungar. Acad. Sci., Budapest, 2002, available at http://www.math-inst.hu/pub/algebraic-logic/Contents.html.
  3. Andréka H., Madarász J.X., Németi I., Székely G., Axiomatizing relativistic dynamics without conservation postulates, Studia Logica 89 (2008), 163-186, arXiv:0801.4870.
  4. Arntzenius F., Causal paradoxes in special relativity, British J. Philos. Sci. 41 (1990), 223-243.
  5. Bilaniuk O.M.P., Deshpande V.K., Sudarshan E.C.G., "Meta" relativity, Amer. J. Phys. 30 (1962), 718-723.
  6. Chashchina O.I., Silagadze Z.K., Breaking the light speed barrier, Acta Phys. Polon. B 43 (2012), 1917-1952, arXiv:1112.4714.
  7. d'Inverno R., Introducing Einstein's relativity, Oxford University Press, New York, 1992.
  8. Feinberg G., Possibility of faster-than-light particles, Phys. Rev. 159 (1967), 1089-1105.
  9. Feynman R.P., The theory of positrons, Phys. Rev. 76 (1949), 749-759.
  10. Firk F.W.K., Introduction to relativistic collisions, arXiv:1011.1943.
  11. Geroch R., Faster than light?, in Advances in Lorentzian Geometry, AMS/IP Stud. Adv. Math., Vol. 49, Amer. Math. Soc., Providence, RI, 2011, 59-69, arXiv:1005.1614.
  12. Jentschura U.D., Wundt B.J., Neutrino helicity reversal and fundamental symmetries, arXiv:1206.6342.
  13. Kleppner D., Kolenkow R.J., An introduction to mechanics, Cambridge University Press, Cambridge, 2010.
  14. Longhi S., Laporta P., Belmonte M., Recami E., Measurement of superluminal optical tunneling times in double-barrier photonic band gaps, Phys. Rev. E 65 (2002), 046610, 6 pages, physics/0201013.
  15. Madarász J.X., Székely G., The existence of superluminal particles is consistent with relativistic dynamics, arXiv:1303.0399.
  16. Nikolić H., Causal paradoxes: a conflict between relativity and the arrow of time, Found. Phys. Lett. 19 (2006), 259-267, gr-qc/0403121.
  17. Nimtz G., Heitmann W., Superluminal photonic tunneling and quantum electronics, Progr. Quantum Electron. 21 (1997), 81-108.
  18. Olkhovsky V.S., Recami E., Jakiel J., Unified time analysis of photon and particle tunnelling, Phys. Rep. 398 (2004), 133-178.
  19. Peacock K.A., Would superluminal influences violate the principle of relativity?, arXiv:1301.0307.
  20. Ranfagni A., Fabeni P., Pazzi G.P., Mugnai D., Anomalous pulse delay in microwave propagation: a plausible connection to the tunneling time, Phys. Rev. E 48 (1993), 1453-1460.
  21. Recami E., Classical tachyons and possible applications, Riv. Nuovo Cimento 9 (1986), 1-178.
  22. Recami E., Tachyon kinematics and causality: a systematic thorough analysis of the tachyon causal paradoxes, Found. Phys. 17 (1987), 239-296.
  23. Recami E., Superluminal tunnelling through successive barriers: does QM predict infinite group-velocities?, J. Modern Opt. 51 (2004), 913-923.
  24. Recami E., A homage to E.C.G. Sudarshan: superluminal objects and waves (an updated overview of the relevant experiments), arXiv:0804.1502.
  25. Recami E., The Tolman-Regge antitelephone paradox: its solution by tachyon mechanics, Electron. J. Theor. Phys. 6 (2009), 8 pages.
  26. Recami E., Zamboni-Rached M., Dartora C.A., Localized X-shaped field generated by a superluminal electric charge, Phys. Rev. E 69 (2004), 027602, 4 pages.
  27. Rindler W., Relativity. Special, general, and cosmological, 2nd ed., Oxford University Press, New York, 2006.
  28. Stannett M., Németi I., Using Isabelle/HOL to verify first-order relativity theory, J. Autom. Reasoning, to appear, arXiv:1211.6468.
  29. Steinberg A.M., Kwiat P.G., Chiao R.Y., Measurement of the single-photon tunneling time, Phys. Rev. Lett. 71 (1993), 708-711.
  30. Stückelberg E.C.G., Un nouveau modèle de l'électron ponctuel en théorie classique, Helv. Phys. Acta 14 (1941), 51-80.
  31. Sudarshan E.C.G., The theory of particles traveling faster than light. I, in Symposia on Theoretical Physics and Mathematics (Madras, India), Editor A. Ramakrishnan, Plenum Press, New York, 1970, 129-151.
  32. Székely G., On why-questions in physics, in The Vienna Circle in Hungary, Editors A. Máté, M. Rédei, F. Stadler, Springer-Verlag, Wien, 2011, 181-189, arXiv:1101.4281.
  33. Tolman R.C., The theory of the relativity of motion, University of California, Berkeley, 1917.
  34. Zamboni-Rached M., Recami E., Besieris I.M., Cherenkov radiation versus X-shaped localized waves, J. Opt. Soc. Amer. A 27 (2010), 928-934.
  35. Zamboni-Rached M., Recami E., Besieris I.M., Cherenkov radiation versus X-shaped localized waves: reply, J. Opt. Soc. Amer. A 29 (2012), 2536-2541.

Previous article  Next article   Contents of Volume 10 (2014)